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  • Mathematical Reasoning

    Focus of Competency 2

    Direction de la formation générale des jeunes Secteur de l’éducation préscolaire et de l’enseignement primaire et secondaire

    Ministère de l’Éducation et de l’Enseignement supérieur

  • Focus of the Competency

    Using mathematical reasoning involves making

    conjectures and criticizing, justifying or refuting a

    proposition by applying an organized body of

    mathematical knowledge.

    Québec Education Program (QEP), Secondary Cycle One, p. 200.

  • By the end of Secondary Cycle One, students should be able to . . .

    • Define a situation and propose conjectures.

    • Apply concepts and processes appropriate to the situation.

    • Try different approaches in order to determine whether they should confirm or refute their conjectures. They validate them either by basing each step of their solution on concepts, processes, rules or statements that they express in an organized manner, or by supplying counterexamples.

    QEP, Cycle One, p. 203.

    Statements considered to be true even though they have not been proven

    Reject the statement

  • By the end of Secondary Cycle two, students in all three options should be able to . . .

    • make conjectures, apply appropriate concepts and processes and confirm or refute their conjectures by using various types of reasoning

    • validate conjectures by basing each step in their proof on concepts, processes, rules or postulates, which they express in an organized manner

    QEP, Secondary Cycle Two, Mathematics, p. 30. Set of justifications based on

    observations, definitions and theorems

    Reject the statement

  • Main Types of Reasoning

    Analogical reasoning

    specific to each branch of

    mathematics

    Inductive reasoning Deductive

    reasoning

    Refutation using counterexamples

    The types of

    reasoning

    specific to each

    branch of

    mathematics are arithmetic,

    proportional,

    algebraic,

    geometric,

    probabilistic and

    statistical

    reasoning.

  • Inductive Reasoning

    Inductive reasoning involves

    generalizing on the basis of

    individual cases.

    QEP, Cycle Two, p. 26.

  • Analogical reasoning

    Analogical reasoning involves

    making comparisons based on

    similarities in order to draw

    conclusions [or make conjectures].

    QEP, Cycle Two, p. 26.

  • Deductive Reasoning

    Deductive reasoning involves a

    [logical] series of propositions that

    lead to conclusions based on

    principles that are considered to be

    true.

    QEP, Cycle Two, p. 26.

  • ➔ Only one counterexample is required to show that a conjecture is false.

    ➔ One cannot conclude that a mathematical statement is true simply because several

    examples show it to be true. QEP, Cycle One, p. 201.

    Refutation Using Counterexamples

    Refutation using counterexamples

    involves disproving a conjecture without

    stating what is true.

    QEP, Cycle Two, p. 26.

  • Main Types of Reasoning

    Analogical reasoning

    specific to each branch of mathematics

    Inductive reasoning Deductive

    reasoning

    Refutation using counterexamples

  • Strategies for Exercising the Competencies

    • Representing the situation mentally or in writing • Giving examples • Finding patterns • Anticipating and interpreting results in light of the context • Referring to a similar problem that has already been solved • Deriving new data from known data

    QEP, Cycle One, p. 220. QEP, Cycle Two, p. 111-112.

  • Strategies for Exercising the Competencies

    • Comparing and questioning one’s procedures and results with those of the teacher or one’s peers

    • Understanding definitions, properties and theorems in order to use them in other contexts

    • Analyzing examples; making comparisons; identifying differences and similarities

    • Assessing the relevance of qualitative or quantitative data • Etc.

    QEP, Cycle One, p. 220. QEP, Cycle Two, p. 111-112.

  • Reasoning and Learning

  • Distinction

    Calculate the area of a circle whose radius is:

    a) 3 cm b) 6 cm

    Exercise involving applications

    What happens to the area of a circle if its radius is doubled?

    Reasoning task

  • Area = 2.4 cm2

    2 cm

    1 .2 cm

    2 .4 cm

    2 cm

    Area = 4.8 cm2

    Are the students’ results consistent with the proposed

    relationship?

    Do all these examples provide sufficient justification for stating

    that if the height doubles, the area also doubles?

    Example of preparatory questions

    What happens to the area of a rectangle if its height is doubled?

  • The area of a rectangle can be determined by multiplying the measure of its base by its height (A = b × h).

    If the initial height is doubled, the area of the new rectangle is determined by multiplying the measure of its base, which is still the same, by the new

    height, which is the initial height multiplied by 2.

    This is equivalent to multiplying the area of the initial rectangle by 2, which is why the area will be twice as large.

    Example of a justification to accompany the conjecture

    What happens to the area of a circle if its radius is doubled?

  • Confirm or refute the following statement: When the radius of a circle is doubled, the area of the circle also doubles.

    What happens to the area of a circle if its radius is doubled?

    Various formulations

  • EXAMPLES OF SITUATIONS

  • Examples: Cycle One

    ➔ Describe what happens to the perimeter of a rectangle when its

    dimensions are doubled, tripled or quadrupled.

    ➔ How is it possible to obtain a unit fraction by subtracting one unit

    fraction from another unit fraction?

    ➔ In the Cartesian plane, what is the geometric relationship between

    the points whose x-coordinate and y-coordinate add up to 5?

  • Examples: Cycle One (cont.)

    ➔ Show that the sum of the measures of the exterior angles of a

    triangle is 360o.

    ➔ Is the following statement true or false?

    In a statistical distribution, when the value of each data item is

    doubled, the mean also doubles.

    Prove by using rigorous reasoning based on properties, definitions and justifications

  • Examples: Cycle One (cont.)

    ➔ Confirm or refute the following statement: When two opposite

    numbers are added to a statistical distribution, the mean does not

    change.

    ➔ A hamburger consists of bread, tomatoes, lettuce and meat. If the

    price of each of these ingredients increases by 5%, by what

    percentage will the total price of the hamburger increase?

    Use a proof to verify that the statement is true

    Find a counterexample

  • ➔ Choose two integers greater than zero. Then, determine their greatest common divisor (GCD) and their least common multiple (LCM). What can you say about the product of the GCD and the LCM of these two numbers?

    ➔ How many solutions does the equation 𝑥2 = 36 have?

    Examples: Cycle One (cont.)

  • Examples: Secondary III

    ➔ Is the following statement true or false? More teams of 3 people

    than teams of 9 people can be formed from a group of 12 people.

    ➔ Show that the expressions and are

    equivalent if .

  • Examples: Secondary III (cont. )

    ➔ Confirm or refute the following statement: When the water in a

    cylindrical container is emptied at a constant rate, the relationship

    between the height of the water in the cylinder and the remaining

    volume of water corresponds to a first-degree function.

    ➔ In a right triangle, an altitude is drawn from the vertex of the right

    angle. What is the relationship between the lengths of the legs, the

    length of the hypotenuse and the length of the altitude drawn?

    Explain.

  • Example: Secondary III (cont.)

    ➔ Why does the direction of the inequality symbol (,≤ and ≥) change when the terms of an inequality are multiplied or divided by a negative number?

  • Examples: Secondary IV

    ➔ What conjecture can you make concerning the sine of two

    supplementary angles?

    ➔ Is the following statement true or false? All inverses of functions

    are functions.

    If the statement is true, provide a proof. If it is false, provide a

    counterexample.

  • Examples: Secondary IV (cont.)

    ➔ Confirm or refute the following statement: Two statistical distributions with the same mean deviation have the same mean.

    ➔ Prove the following statements: ● In a circle, two congruent central angles have congruent chords. ● Any straight line that intersects two sides of a triangle and is parallel to the

    third side divides the two sides into segments of proportional le

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